Date: 05/23/2003 at 04:26:49 From: Marc Subject: Congruence symbol How does congruence (3-lined symbol) relate to equality (=)? When I say two equations are congruent, what am I actually saying? e.g. when I say two equations are equal, I mean that they represent the same numerical value, no? Or perhaps that for all possible values of their variables they are equal? Thanks.
Date: 05/23/2003 at 12:51:51 From: Doctor Peterson Subject: Re: Congruence symbol Hi, Marc. The "triple-equal" symbol (which I will represent here with "==") is actually used in several different ways, one of which you appear to be using, but with the name given to a different usage. First, please note that you are referring to the symbol being used not between two EQUATIONS, but two EXPRESSIONS. That makes a big difference here; an equation includes an equals sign. One use is for an "is defined as": Defined - Eric Weisstein's World of Mathematics http://mathworld.wolfram.com/Defined.html We can say, for example, that tan(x) == sin(x)/cos(x) to say that the tangent is defined as the ratio of sine to cosine, if we choose to define it that way. A closely related use is for "identical equality," which I think is what you have in mind: Identities - AS Guru Maths (BBCi) http://www.bbc.co.uk/education/asguru/maths/13pure/04trigonometry/ 23identities/index.shtml This indicates an "identity," an equation that is always true, for any value of its variable(s). This is sometimes denoted in the same way, and read "is identically equal to": (a + b)^2 == a^2 + 2ab + b^2 because this is true for all a and b. The important point is that it is not just its numerical value that happens to be the same in one instance, but rather the meaning of one expression, or function, itself is exactly the same as the other. That makes this a deeper level of equality. A third use is for "congruence" in number theory: Congruence - Eric Weisstein's World of Mathematics http://mathworld.wolfram.com/Congruence.html We say that two numbers are "congruent modulo n" or a == b (mod n) if a - b is a multiple of n. Another use, which I am not familiar with, is in logic: Equivalent - Eric Weisstein's World of Mathematics http://mathworld.wolfram.com/Equivalent.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 10/14/2004 at 12:51:51 From: Cristian Subject: Congruence symbol There is another use of the '==' symbol: in geometry, where two shapes are congruent, they are numerically (i.e., except for positioning) equal. -Cristian
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